For those not familiar with Sawyer's other books (*Prelude to Mathematics* -- also by Dover Press, and *Mathematician's Delight* -- by Penguin, but out-of-print), W. W. Sawyer was a Brit who excelled in making even the most opaque topic in mathematics understandable and clear to his many students and readers.

Sawyer's *Vision in Elementary Mathematics* adds to his sterling reputation. On the surface, *Vision* focuses on various topics in 'elementary' mathematics including: algebra, arithmetic, and geometry. A deeper look at this book reveals, however, that Sawyer's book goes well beyond rehashing basic concepts in math.

The focus in this book -- as the title *Vision* suggests -- is on helping both students and teachers have a stronger inuitive grasp of basic concepts of mathematics that many teachers tend to gloss over. As many people have experienced in their own education, mathematics is often treated as a mysterious, 'black box'-like subject. That kind of mindless and unthinking approach to mathematics teaching and learning tend to turn many people off to the subject. This ongoing tragedy in American education has been well researched by another excellent book, *Knowing and Teaching Elementary Mathematics* by Liping Ma.

Sawyer wrote this book in order to combat the unthinking approach to math education. The remedy he offers to that kind of approach is to encourage a deeper understanding of mathematics from relatively simple concepts like fractions, arithmetic, and number properties all the way up to polynomial equations. My favorite quote from this book, which is repeated in different forms, is: "We first try to make sure that we can see what the problem means - for if we do not understand the question, we have no hope of finding the answer." As this quote suggests, *Vision* focuses on helping people to truly UNDERSTAND what mathematical problems really mean.

*Vision* utilizes various techniques to help readers to have a deeper level of understanding of mathematics. Sawyer encourages people to develop some geometrical heuristics to help them grasp a problem. A great example of this is how Sawyer explains how to easily multiply and divide anything from numbers all the way up to polynomial algebraic equations using 'matrix'-like geometric concepts -- no mean feat!

Another great thing about *Vision* is that Sawyer encourages experimentation by students in this book. As many math-lovers know, you can't really learn mathematics without doing it. Sawyer not only encourages people to 'do' math but to experiment with it. By encouraging students to think about math problems in creative and novel ways -- and to not be afraid of making mistakes and, hopefully, learning from them -- *Vision* will help many math students to get a greater appreciation for the subject and will encourage them to be much more creative about the subject. Furthermore, by encouraging student to find out for themselves the whys and hows of a correct solution, *Vision* will encourage the kind of mentality one has to have to become a good mathematician or scientist.

In this regard, it should also be noted that Sawyer includes several exercises at the end of each chapter and includes ANSWERS at the end of the book for each and every question (very helpful for self-learning).

A surprising -- given its focus on 'elementary' mathematics -- benefit of this book is that it provides solid foundations for more advance topics in mathematics. *Vision* provides excellent foundations for number theory, linear algebra, calculus, trigonometry, and combinatorics. Another plus for this book is that Sawyer deliberately tries to emphasize the fact that mathematics -- at any level -- can be useful to real-life problems because mathematics underlies so much of what we see in the natural world.

Who should read this book? The obvious audiences are teachers and students of mathematics in anywhere from elementary school up to high school. This book would be especially helpful to home-schoolers who are interested in providing their kids with a solid grounding in mathematics that can be built upon when they attend university.

But those aren't the only people who should read this book. This book should be read by people who feel that they had an 'incomplete' education. People, like myself, who felt a bit cheated by the poor state of the educational system should definitely read this book no matter how confident (or un-confident) about their mathematical abilities.
This book can and should be read by advance students and teachers of mathematics in fields ranging from engineers to scientists (including professional mathematicians). Those advanced users of mathematics can use this book both for review and to get some additional insights into their chosen subject. I have no doubt that Sawyer has insights and creative approaches in *Vision* that will prove valuable to mathematicians at any level.

Finally, ANYONE interested in getting a 'profound understanding of fundamental mathematics' (often abbreviated PUFM in educational policy circles) for whatever reason MUST read this wonderful book. *Vision* perfectly fulfills Sawyer's vision of writing a book that helps people to look at mathematics in a novel way that will help them to have a deeper understanding and appreciation for mathematics.

I'm surprised that I'm the first person to review this great book by W. W. Sawyer. For those not familiar with Sawyer's other books (*Prelude to Mathematics* -- also by Dover Press, and *Mathematician's Delight* -- by Penguin, but out-of-print), W. W. Sawyer was a Brit who excelled in making even the most opaque topic in mathematics understandable and clear to his many students and readers.

Sawyer's *Vision in Elementary Mathematics* adds to his sterling reputation. On the surface, *Vision* focuses on various topics in 'elementary' mathematics including: algebra, arithmetic, and geometry. A deeper look at this book reveals, however, that Sawyer's book goes well beyond rehashing basic concepts in math.

The focus in this book -- as the title *Vision* suggests -- is on helping both students and teachers have a stronger inuitive grasp of basic concepts of mathematics that many teachers tend to gloss over. As many people have experienced in their own education, mathematics is often treated as a mysterious, 'black box'-like subject. That kind of mindless and unthinking approach to mathematics teaching and learning tend to turn many people off to the subject. This ongoing tragedy in American education has been well researched by another excellent book, *Knowing and Teaching Elementary Mathematics* by Liping Ma.

Sawyer wrote this book in order to combat the unthinking approach to math education. The remedy he offers to that kind of approach is to encourage a deeper understanding of mathematics from relatively simple concepts like fractions, arithmetic, and number properties all the way up to polynomial equations. My favorite quote from this book, which is repeated in different forms, is: "We first try to make sure that we can see what the problem means - for if we do not understand the question, we have no hope of finding the answer." As this quote suggests, *Vision* focuses on helping people to truly UNDERSTAND what mathematical problems really mean.

*Vision* utilizes various techniques to help readers to have a deeper level of understanding of mathematics. Sawyer encourages people to develop some geometrical heuristics to help them grasp a problem. A great example of this is how Sawyer explains how to easily multiply and divide anything from numbers all the way up to polynomial algebraic equations using 'matrix'-like geometric concepts -- no mean feat!

Another great thing about *Vision* is that Sawyer encourages experimentation by students in this book. As many math-lovers know, you can't really learn mathematics without doing it. Sawyer not only encourages people to 'do' math but to experiment with it. By encouraging students to think about math problems in creative and novel ways -- and to not be afraid of making mistakes and, hopefully, learning from them -- *Vision* will help many math students to get a greater appreciation for the subject and will encourage them to be much more creative about the subject. Furthermore, by encouraging student to find out for themselves the whys and hows of a correct solution, *Vision* will encourage the kind of mentality one has to have to become a good mathematician or scientist.

In this regard, it should also be noted that Sawyer includes several exercises at the end of each chapter and includes ANSWERS at the end of the book for each and every question (very helpful for self-learning).

A surprising -- given its focus on 'elementary' mathematics -- benefit of this book is that it provides solid foundations for more advance topics in mathematics. *Vision* provides excellent foundations for number theory, linear algebra, calculus, trigonometry, and combinatorics. Another plus for this book is that Sawyer deliberately tries to emphasize the fact that mathematics -- at any level -- can be useful to real-life problems because mathematics underlies so much of what we see in the natural world.

Who should read this book? The obvious audiences are teachers and students of mathematics in anywhere from elementary school up to high school. This book would be especially helpful to home-schoolers who are interested in providing their kids with a solid grounding in mathematics that can be built upon when they attend university.

But those aren't the only people who should read this book. This book should be read by people who feel that they had an 'incomplete' education. People, like myself, who felt a bit cheated by the poor state of the educational system should definitely read this book no matter how confident (or un-confident) about their mathematical abilities.

This book can and should be read by advance students and teachers of mathematics in fields ranging from engineers to scientists (including professional mathematicians). Those advanced users of mathematics can use this book both for review and to get some additional insights into their chosen subject. I have no doubt that Sawyer has insights and creative approaches in *Vision* that will prove valuable to mathematicians at any level.

Finally, ANYONE interested in getting a 'profound understanding of fundamental mathematics' (often abbreviated PUFM in educational policy circles) for whatever reason MUST read this wonderful book. *Vision* perfectly fulfills Sawyer's vision of writing a book that helps people to look at mathematics in a novel way that will help them to have a deeper understanding and appreciation for mathematics.

Whenever I teach calculus, my emphasis is always on the fact that the basic ideas of calculus can be understood by almost anyone. While I am not always successful in proving this to the students, Sawyer certainly would be. His explanations of the basics of a derivative are the clearest, most understandable that I have ever seen. There are many diagrams, and each one has a specific purpose and they are well integrated into the textual explanations.
This is not a book that you could use to teach a college calculus course. The mandatory epsilons and deltas that form the backbone of basic calculus are mentioned only as an incidental. Sawyer sets out to explain the foundation ideas of calculus in terms of everyday occurrences and for that reason it is better suited to someone who is curious about calculus. However, it could be used as a supplemental text in the foundations of science, as calculus is used in all areas of change, which describes almost all of nature.
While the notation of mathematics is concise, abstract and often appears esoteric, many of the ideas expressed in that notation are quite easy to follow. In this book, Sawyer explains what calculus is all about in terms that anyone who understands motion can follow. There needs to be more people like him writing books like this.

When I was reading this book all along, I felt comfortable. I heard it as if my grandfather told an interesting story. When the story was over, I became a real university student. I had learned a vacant caculus, but I have known that the calculus 'is' our life.

Its about a boy named Tom Sawyer who is the town tricker he always has a trick up his sleeve but all the tricks stop when Tom and Huck see Injuin Joe murdering Doc. Robinson in the graveyard .It was all fun and jokes until that night, then things got serious.

tom sawyer lives with his aunt polly in St Petersburg,a town on the Mississippi.

Frequently forgotten or misjudged, this book provides one of the most hilarious plots ever written. The "common sense" of Huck is always fighting for bring Tom's lunacies to Earth, but the question is, why should we do that? Tom's reveries are the key for happy travel with a sane state of mind, providing the key which erase the nuance between what should be and possibly, what actually is. A delightful book.

Laura Huxley herself said to Sawyer [paraphrased], "Out of all the biographies written about Aldous, this is the only one he would have actually liked." With emphasis on philosophical studies and works, it is the definitive source for understanding Huxley's influence and ideas. Being a student myself at Maine College of Art, I can vouch for the know-how of Sawyer.

A fascinating inside look at President Truman's seizure of the steel mills during the Korean War and the constitutional crisis that followed. Westin pieces together bits and pieces of key legal briefs, Truman's memoirs, and an account of the social and political forces operating at the time of the seizure. If you ever wanted to know what the key actors were really thinking and what happened after the Supreme Court declared Truman's action unconstitutional, this book is a must read.

This is an excellent book for 4-12th grades. It has interesting writing and project ideas to stimulate creative learning. It would also be a valuable resource for homeschool parents teaching on the subject of the Ancient World.

This is one book which an engineer esp an environemtal engineer must have as it starts from basics and takes you to complex situations making them understandable.

sorry, I don't speakg english very well, I only try . I need buy this book, that was indicated by a friend, but I have a problem(I never bought a book in another country e I don't have credit card). Please help me.